Hi Troy,

Sorry about that. I did not realize you were looking for a generic solution. Regarding

how to pattern match the sequence of Power[z1,1/2]Power[z2,1/2]

Power[z1, 1/2] Power[z2, 1/2] /. 
 Times[Power[a_, 1/2], Power[b_, 1/2]] :> Power[a b, 1/2]
(* Sqrt[z1 z2] *)

But again, that might just be a specific solution and potentially dangerous since it is not true for all z1 and z2.

Hi Troy,

You need to provide some assumptions for that identity to be true.

exp = Sqrt[a + b]*Sqrt[a - b]
Simplify[exp, a + b >= 0 && a - b >= 0]
(* Sqrt[a^2 - b^2] *)